Rough differential equations and planarly branched universal limit theorem
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| Format: | Preprint |
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2024
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| _version_ | 1866911008075808768 |
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| author | Gao, Xing Li, Nannan Manchon, Dominique |
| author_facet | Gao, Xing Li, Nannan Manchon, Dominique |
| contents | The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness $\frac{1}{3}< α\leq \frac{1}{2}$; (ii) geometric rough paths with roughness $0< α\leq 1$; (iii) branched rough paths with roughness $0< α\leq 1$. Planarly branched rough paths are natural generalizations of both rough paths and branched rough paths, in the sense that post-Lie algebras are generalizations of both Lie algebras and pre-Lie algebras. Here the primitive elements of the graded dual Hopf algebra of the Hopf algebra corresponding to the planarly branched rough paths (resp. rough paths, resp. branched rough paths) form a post-Lie (resp. Lie, resp. pre-Lie algebra). In this paper, we prove the universal limit theorem for planarly branched rough paths with roughness $\frac{1}{4}< α\leq \frac{1}{3}$, via the method of Banach fixed point theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rough differential equations and planarly branched universal limit theorem Gao, Xing Li, Nannan Manchon, Dominique Probability Classical Analysis and ODEs 60L20, 60L50, 60H99, 34K50, 37H10, 05C05 The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness $\frac{1}{3}< α\leq \frac{1}{2}$; (ii) geometric rough paths with roughness $0< α\leq 1$; (iii) branched rough paths with roughness $0< α\leq 1$. Planarly branched rough paths are natural generalizations of both rough paths and branched rough paths, in the sense that post-Lie algebras are generalizations of both Lie algebras and pre-Lie algebras. Here the primitive elements of the graded dual Hopf algebra of the Hopf algebra corresponding to the planarly branched rough paths (resp. rough paths, resp. branched rough paths) form a post-Lie (resp. Lie, resp. pre-Lie algebra). In this paper, we prove the universal limit theorem for planarly branched rough paths with roughness $\frac{1}{4}< α\leq \frac{1}{3}$, via the method of Banach fixed point theorem. |
| title | Rough differential equations and planarly branched universal limit theorem |
| topic | Probability Classical Analysis and ODEs 60L20, 60L50, 60H99, 34K50, 37H10, 05C05 |
| url | https://arxiv.org/abs/2412.16479 |